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Quantum Criticality

HARVARD

Talk online: sachdev.physics.harvard.edu

S. Sachdev and B. Keimer, Physics Today, February 2011

Thursday, May 5, 2011

What is a quantum phase transition ?

Non-analyticity in ground state properties as a function of some control parameter g




E

g

True level crossing:

Usually a first-order transition




E

g

True level crossing:

Usually a first-order transition

E

g

Avoided level crossing which becomes sharp in the infinite

volume limit:

second-order transitionThursday, May 5, 2011

g

Many levels are important near a second-order quantum phase transition


Why study quantum phase transitions ?

ggc

• The ground states at large and small g have wavefunctions which can usually be written as products of wavefunctions of local degrees of freedom i.e. they have negligible quantum entanglement. The quantum critical state at gc often has long-range quantum entanglement : the “spooky” non-local quantum correlations pointed out by Einstein, Podolsky, and Rosen survive in a macrosopic system at the longest distances.



ggc

• We are often able to describe the quantum state at gc by methods drawn from quantum field theory; expansion in g-gc then allows for a controlled theory in an intermediate coupling regime important for many experimental systems


T Quantum criticality


ggc

• The quantum critical point controls properties over a wide regime of “quantum criticality” at non-zero temperatures. I will argue that this regime is the key to understanding the physical properties of a variety of modern electronic materials.


1. The quantum Ising chain

A. The magnetic insulator CoNb2O6

B. Ultracold Rb atoms in an optical lattice

2. Nonzero temperatures and quantum criticality Antiferromagnetic insulators

3. Higher temperature superconductors and “strange metals” Quantum criticality of fermions and Fermi surfaces

Outline


( ) ( )

Degrees of freedom: 1 qubits, "large"

,

1 1 or ,

2 2

j j

j jj j j j

j N N=

=!

"

"

#

# #+ =$ % "

!...


( ) ( )

Degrees of freedom: 1 qubits, "large"

,

1 1 or ,

2 2

j j

j jj j j j

j N N=

=!

"

"

#

# #+ =$ % "

!...

0

Hamiltonian of decoupled qubits:

xj

j

H Jg != " # 2Jg

j!

j!


1 1

Coupling between qubits:

z zj j

j

H J ! ! += " #


1 1


z zj j

j

H J ! ! += " #

( )( )1 1j j j j+ ++ +! !" !" "! "


1 1


z zj j

j

H J ! ! += " #

1 1

Prefers neighboring qubits

are

(not entangle

d)j j j j

either or+ +

! !" "


( )0 1 1x z zj j j

j

J gH H H ! ! ! += + = " +#

Full Hamiltonian

ggc


( )0 1 1x z zj j j

j

J gH H H ! ! ! += + = " +#

Full Hamiltonian

ggc

|→�1 |→�2 . . . |→�j . . . |→�N−1 |→�N

Product state for large gThursday, May 5, 2011

( )0 1 1x z zj j j

j

J gH H H ! ! ! += + = " +#

Full Hamiltonian

ggc

Product state for small g

|↑�1 |↑�2 . . . |↑�j . . . |↑�N−1 |↑�N|↓�1 |↓�2 . . . |↓�j . . . |↓�N−1 |↓�N

or


( )0 1 1x z zj j j

j

J gH H H ! ! ! += + = " +#

Full Hamiltonian

ggc

Entangled state at quantum critical point, involving complicated superposition of 2N

qubit configurations







Outline


R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran,

M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177 (2010).

Quasi-1D Ising ferromagnet CoNb2O6

Single crystal of CoNb2O6

(Oxford image

furnace)

4 cm

c

Co2+

cb

a

Co2+ spin chain along c-axis

CoO6distorted

octahedron

Co2+

Oxygen

Ferromagnetic superexchange

~ 90° bond Co-O-Co ~ 20K ~ 2meV

Strong easy-axis

(Ising)

…

30 meV


R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran,

M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177 (2010).

Quasi-1D Ising ferromagnet CoNb2O6

c

Co2+

cb

a

Co2+ spin chain along c-axis Magnetic long-range order

Bragg peak

Transverse field Jg (Tesla)







Outline


M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Superfluid-insulator transition of 87Rb atoms in a magnetic trap and an optical lattice potential


M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Mott insulator of 87Rb atoms in a magnetic trap and an optical lattice potential


Applying an “electric” field to the Mott insulator


Why is there a peak (and

not a threshold)

when E = U ?


Resonant transition when E≈U


Virtual state


Resonant transition when E≈U


Phase diagram

(E-U)/t

S. Sachdev, K. Sengupta, and S.M. Girvin, Phys. Rev. B 66, 075128 (2002)


(E-U)/tIsing quantum

phase transition

S. Sachdev, K. Sengupta, and S.M. Girvin, Phys. Rev. B 66, 075128 (2002)

Phase diagram


Effective Hamiltonian can be written as spin model

= x-y plane

= up

= down

Hamiltonian of resonant subspace



= x-y plane

= up

= down

forbidden !

Constraint:



Include a term (J/4)�

i (σzi − 1)

�σzi+1 − 1

�

and send J → ∞. Infinite exchange interaction !


= x-y plane

= up

= down

forbidden !

Constraint:




Δ=E-U <0

Paramagnetic state




Δ=E-U > 0

Antiferromagnetic state, two fold degenerate



Antiferromagnetic state, two fold degenerate

+



H =�

i

�J

4σzi σ

zi+1

− hz

2σzi − hx

2σxi

�

J → ∞,

hz = J + (U − E),

hx = 2√2t

hx =0: classical first order phase transitionFinite hx: quantum phase transition, second order

Phase diagram of spin model

hz/J

hx/J


J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss,and M. Greiner, Nature 472, 307 (2011)


Quantum gas microscope



In-situ imaging of antiferromagnetic chains




=

=




=

=

Antiferromagnetic order

= 1/gJ. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss,and M. Greiner, Nature 472, 307 (2011)



Hanbury-Brown-Twiss noise correlations measureFourier transform of boson density



Hanbury-Brown-Twiss noise correlations measureFourier transform of boson density

Peaks fromantiferromagnetic order







Outline


The cuprate superconductors


Ground state has long-range Néel order

Square lattice antiferromagnet

H =�

�ij�

Jij�Si · �Sj

Order parameter is a single vector field �ϕ = ηi�Si

ηi = ±1 on two sublattices

��ϕ� �= 0 in Neel state.



H =�

�ij�

Jij�Si · �Sj

J

J/λ

Weaken some bonds to induce spin entanglement in a new quantum phase



H =�

�ij�

Jij�Si · �Sj

J

J/λ

Ground state is a “quantum paramagnet”with spins locked in valence bond singlets

=1√2

��↑↓�−

�� ↓↑��


λλc

=1√2

��↑↓�−

�� ↓↑��


Pressure in TlCuCl3

λλc

=1√2

��↑↓�−

�� ↓↑��

A. Oosawa, K. Kakurai, T. Osakabe, M. Nakamura, M. Takeda, and H. Tanaka, Journal of the Physical Society of Japan, 73, 1446 (2004).


TlCuCl3

An insulator whose spin susceptibility vanishes exponentially as the temperature T tends to zero.


TlCuCl3

Quantum paramagnet at ambient pressure


TlCuCl3

Neel order under pressureA. Oosawa, K. Kakurai, T. Osakabe, M. Nakamura, M. Takeda, and H. Tanaka, Journal of the Physical Society of Japan, 73, 1446 (2004).


λλc

=1√2

��↑↓�−

�� ↓↑��


λλcSpin S = 1“triplon”

Excitation spectrum in the paramagnetic phase


λλc

Excitation spectrum in the paramagnetic phase

Spin S = 1“triplon”


λλc

Excitation spectrum in the Neel phase

Spin waves


Triplon in the quantum paramagnet.

Spin waves and a new “Higgs” particle in the Neel phase: the latter

represents longitudinal oscillations in the magnitude of the Neel order.

Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans–Ulrich Gudel, Severian Gvasaliya,

Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, 205701 (2008)

TlCuCl3 with varying pressure

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

Pressure [kbar]

Ener

gy [m

eV]








0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

Pressure [kbar]

Ener

gy [m

eV]

Triplon energy gap∆ ∼ (λ− λc)zν








0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

Pressure [kbar]

Ener

gy [m

eV]

Spin waves








0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

Pressure [kbar]

Ener

gy [m

eV]

“Higgs”








0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

Pressure [kbar]

Ener

gy [m

eV]


λλc

Quantum critical point with non-local entanglement in spin wavefunction

=1√2

��↑↓�−

�� ↓↑��


Classicalspin

waves

Dilutetriplon

gas

Quantumcritical

Neel orderPressure in TlCuCl3

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, 11919 (1994).


Classicalspin

waves

Dilutetriplon

gas

Quantumcritical



Triplon energy gap∆ ∼ (λ− λc)zν ;crossover at∆ ∼ kBT .


Classicalspin

waves

Dilutetriplon

gas

Quantumcritical



Classical Boltzmann theory oftriplon particles:

Leads to relaxation andthermal equilibration timesof order (�/kBT )e∆/kBT


Neutron scattering measurements of the collisions between

triplons in Y2BaNiO5

Guangyong Xu, C. Broholm, Yeong-Ah Soh, G. Aeppli, J. F. DiTusa, Ying Chen,M. Kenzelmann, C. D. Frost, T. Ito, K. Oka, H. Takagi, Science 317, 1049 (2007)


Neutron scattering measurements of the collisions between

triplons in Y2BaNiO5

Guangyong Xu, C. Broholm, Yeong-Ah Soh, G. Aeppli, J. F. DiTusa, Ying Chen,M. Kenzelmann, C. D. Frost, T. Ito, K. Oka, H. Takagi, Science 317, 1049 (2007)

Theoretical prediction�Γ = 1.20kBTe∆/kBT

K. Damle and S. Sachdev,Phys. Rev. B 57, 8307 (1998)


Classicalspin

waves

Dilutetriplon

gas

Quantumcritical



Classical Boltzmann theory oftriplon particles:

Leads to relaxation andthermal equilibration timesof order (�/kBT )e∆/kBT


Classicalspin

waves

Dilutetriplon

gas

Quantumcritical



Classical non-linear spinwaves, also withrelaxational and

equilibration times� �/kBT


Classicalspin

waves

Dilutetriplon

gas

Quantumcritical





Classicalspin

waves

Dilutetriplon

gas

Quantumcritical


Strongly coupled dynamics andtransport with no

particle/wave interpretation,and relaxation thermalequilibration times are

universally proportional to�/kBT


Quantum critical transport

S. Sachdev, Quantum Phase Transitions, Cambridge (1999).

Quantum “nearly perfect fluid”with shortest possiblerelaxation time, τR

τR = C �kBT

where C is a universal constant


0

0

Non-zero temperature crossovers for the quantum Ising chain

Color density plot of dξ−1/dT , where ξ is the spin correlation length. This quantitymeasures the strength of the interactions between the thermal excitations.


Neutron scattering measurements on La1.86 Sr0.14 Cu O4, showing scaling of thedynamic spin susceptibility at an incommensurate wavevector:

χ��(ω, T ) =A

T 2−ηΦ

��ωkBT

�

G. Aeppli, T. E. Mason, S. M. Hayden, H. A. Mook and J. Kulda,Science, 278, 1432 (1997).







Outline


The cuprate superconductors


Ground state has long-range Néel order


H =�

�ij�

Jij�Si · �Sj

Order parameter is a single vector field �ϕ = ηi�Si

ηi = ±1 on two sublattices

��ϕ� �= 0 in Neel state.


Hole-doped

Electron-doped


Hole-doped

Electron-doped

Electron-doped cuprate superconductors


Hole-doped

Electron-doped

Resistivity∼ ρ0 +ATn


Figure prepared by K. Jin and and R. L. Greenebased on N. P. Fournier, P. Armitage, and

R. L. Greene, Rev. Mod. Phys. 82, 2421 (2010).


Hole-doped

Electron-doped

Resistivity∼ ρ0 +ATn


Figure prepared by K. Jin and and R. L. Greenebased on N. P. Fournier, P. Armitage, and

R. L. Greene, Rev. Mod. Phys. 82, 2421 (2010).

StrangeMetal


Iron pnictides: a new class of high temperature superconductors


Ishida, Nakai, and HosonoarXiv:0906.2045v1

0 0.02 0.04 0.06 0.08 0.10 0.12

150

100

50

0

SC

Ort

AFM Ort/

Tet

S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, A. I. Goldman,

Physical Review Letters 104, 057006 (2010).

Iron pnictides: a new class of high temperature superconductors


TSDW Tc

T0

2.0

0

!"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

Temperature-doping phase diagram of the iron pnictides:

Resistivity∼ ρ0 +ATα

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B 81, 184519 (2010)

SDW= spin density wave = antiferromagnetism in a metal


TSDW Tc

T0

2.0

0

!"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

Temperature-doping phase diagram of the iron pnictides:


StrangeMetal

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B 81, 184519 (2010)

SDW= spin density wave = antiferromagnetism in a metal


Organic Bechgaard compounds

Doiron-Leyraud et al., PRB 80, 214531 (2009)



Organic Bechgaard compounds

Doiron-Leyraud et al., PRB 80, 214531 (2009)

StrangeMetal



Lower Tc superconductivity in the heavy fermion compounds

G. Knebel, D. Aoki, and J. Flouquet, arXiv:0911.5223

2

Generally, the ground state of a Ce heavy-fermion sys-tem is determined by the competition of the indirect Ru-derman Kittel Kasuya Yosida (RKKY) interaction whichprovokes magnetic order of localized moments mediatedby the light conduction electrons and the Kondo interac-tion. This last local mechanism causes a paramagneticground state due the screening of the local moment ofthe Ce ion by the conduction electrons. Both interac-tions depend critically on the hybridization of the 4felectrons with the conduction electrons. High pressure isan ideal tool to tune the hybridization and the positionof the 4f level with respect to the Fermi level. Thereforehigh pressure experiments are ideal to study the criti-cal region where both interactions are of the same orderand compete. To understand the quantum phase transi-tion from the antiferromagnetic (AF) state to the para-magnetic (PM) state is actually one of the fundamen-tal questions in solid state physics. Di!erent theoreticalapproaches exist to model the magnetic quantum phasetransition such as spin-fluctuation theory of an itinerantmagnet [10–12], or a new so-called ’local’ quantum criti-cal scenario [13, 14]. Another e"cient source to preventlong range antiferromagnetic order is given by the va-lence fluctuations between the trivalent and the tetrava-lent configuration of the cerium ions [15].

The interesting point is that in these strongly corre-lated electron systems the same electrons (or renormal-ized quasiparticles) are responsible for both, magnetismand superconductivity. The above mentioned Ce-115family is an ideal model system, as it allows to studyboth, the quantum critical behavior and the interplay ofthe magnetic order with a superconducting state. Espe-cially, as we will be shown below, unexpected observa-tions will be found, if a magnetic field is applied in thecritical pressure region.

PRESSURE-TEMPERATURE PHASE DIAGRAM

In this article we concentrate on the compoundCeRhIn5. At ambient pressure the RKKY interactionis dominant in CeRhIn5 and magnetic order appears atTN = 3.8 K. However, the ordered magnetic moment ofµ = 0.59µB at 1.9 K is reduced of about 30% in com-parison to that of Ce ion in a crystal field doublet with-out Kondo e!ect [17]. Compared to other heavy fermioncompounds at p = 0 the enhancement of the Sommerfeldcoe"cient of the specific heat (! = 52 mJ mol!1K!2) [18]and the cylotron masses of electrons on the extremal or-bits of the Fermi surface is rather moderate [19, 20]. Thetopologies of the Fermi surfaces of CeRhIn5 are cylin-drical and almost identical to that of LaRhIn5 which isthe non 4f isostructural reference compound. From thisit can be concluded that the 4f electrons in CeRhIn5

are localized and do not contribute to the Fermi volume[19, 20].

By application of pressure, the system can be tunedthrough a quantum phase transition. The Neel temper-

FIG. 2. Pressure–temperature phase diagram of CeRhIn5 atzero magnetic field determined from specific heat measure-ments with antiferromagnetic (AF, blue) and superconduct-ing phases (SC, yellow). When Tc < TN a coexistence phaseAF+SC exist. When Tc > TN the antiferromagnetic order isabruptly suppressed. The blue square indicate the transitionfrom SC to AF+SC after Ref. 16.

ature shows a smooth maximum around 0.8 GPa andis monotonously suppressed for higher pressures. How-ever, CeRhIn5 is also a superconductor in a large pres-sure region from about 1.3 to 5 GPa. It has been shownthat when the superconducting transition temperatureTc > TN the antiferromagnetic order is rapidly sup-pressed (see figure 2) and vanishes at a lower pressurethan that expected from a linear extrapolation to T = 0.Thus the pressure where Tc = TN defines a first criticalpressure p!

c and clearly just above p!c anitferromagnetism

collapses. The intuitive picture is that the opening of asuperconducting gap on large parts of the Fermi surfaceabove p!

c impedes the formation of long range magneticorder. A coexisting phase AF+SC in zero magnetic fieldseems only be formed if on cooling first the magnetic or-der is established. We will discuss below the microscopicevidence of an homogeneous AF+SC phase.

At ambient pressure CeRhIn5 orders in an incommen-surate magnetic structure [21] with an ordering vectorof qic=(0.5, 0.5, ") and " = 0.297 that is a magneticstructure with a di!erent periodicity than the one of thelattice. Generally, an incommensurate magnetic struc-ture is not favorable for superconductivity with d wavesymmetry, which is realized in CeRhIn5 above p!

c [22].Neutron scattering experiments under high pressure donot give conclusive evidence of the structure under pres-sures up to 1.7 GPa which is the highest pressure studiedup to now [23–25]. The result is that at 1.7 GPa the in-commensurability has changed to " ! 0.4. The main dif-ficulty in these experiments with large sample volume isto ensure the pressure homogeneity. Near p!

c the controlof a perfect hydrostaticity is a key issue as the materialreacts quite opposite on uniaxial strain applied along thec and a axis.

From recent nuclear quadrupol resonance (NQR) data


Fermi surface

Metal with “large” Fermi surface

Momenta with electronic

states empty

Momenta with electronic

states occupied


Fermi surface+antiferromagnetism

The electron spin polarization obeys�

�S(r, τ)�

= �ϕ(r, τ)eiK·r

where K is the ordering wavevector.

+



Metal with “large” Fermi surfaceThursday, May 5, 2011

Fermi surfaces translated by K = (π,π).Thursday, May 5, 2011

Electron and hole pockets inantiferromagnetic phase with ��ϕ� �= 0




��ϕ� = 0

Metal with electron and hole pockets

Increasing SDW order

��ϕ� �= 0

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Increasing interaction


Nd2−xCexCuO4

T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner,

M. Lambacher, A. Erb, J. Wosnitza, and R. Gross,

Phys. Rev. Lett. 103, 157002 (2009).

Quantum oscillations


Nd2−xCexCuO4

T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner,

M. Lambacher, A. Erb, J. Wosnitza, and R. Gross,

Phys. Rev. Lett. 103, 157002 (2009).

Quantum oscillations


s




��ϕ� = 0

Metal with electron and hole pockets


��ϕ� �= 0

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

Increasing interaction


FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal

Spin density wave (SDW)

Quantum criticality of antiferromagnetism and Fermi surfaces


T*QuantumCritical


FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal

Spin density wave (SDW)

Quantum criticality of antiferromagnetism and Fermi surfaces


T*QuantumCritical

StrangeMetal


Conclusions

Paradigm of quantum phase transitions:the quantum Ising chain, realized in

the ferromagnetic insulator CoNb2O6, and

ultracold atoms in “tilted” optical lattices


Conclusions

Described excitations spectrum of the coupled-dimer antiferromagnet TlCuCl3.

It has regimes of classical dynamics of triplon particles,

and of non-linear spin waves,and a regime of quantum-criticality with

characteristic equilibration time �/kBT


Conclusions

Quantum criticality of antiferromagnetism

and Fermi surface reconstruction controls “strange metal” regime of

quasi-two dimensional higher temperature superconductors


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